Properties

Label 2.0.4.1-57122.3-h2
Base field \(\Q(\sqrt{-1}) \)
Conductor norm \( 57122 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 1 \)
Rank \( 0 \)

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Base field \(\Q(\sqrt{-1}) \)

Generator \(i\), with minimal polynomial \( x^{2} + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 0, 1]))
 
gp: K = nfinit(Polrev([1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 1]);
 

Weierstrass equation

\({y}^2+i{x}{y}={x}^{3}-764{x}+16264\)
sage: E = EllipticCurve([K([0,1]),K([0,0]),K([0,0]),K([-764,0]),K([16264,0])])
 
gp: E = ellinit([Polrev([0,1]),Polrev([0,0]),Polrev([0,0]),Polrev([-764,0]),Polrev([16264,0])], K);
 
magma: E := EllipticCurve([K![0,1],K![0,0],K![0,0],K![-764,0],K![16264,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((169i+169)\) = \((i+1)\cdot(-3i-2)^{2}\cdot(2i+3)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 57122 \) = \(2\cdot13^{2}\cdot13^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-84835994984)\) = \((i+1)^{6}\cdot(-3i-2)^{9}\cdot(2i+3)^{9}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 7197146044925273160256 \) = \(2^{6}\cdot13^{9}\cdot13^{9}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{10218313}{17576} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \(0\)
Torsion structure: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 0.20698479944204956544811175026614512114 \)
Tamagawa product: \( 24 \)  =  \(( 2 \cdot 3 )\cdot2\cdot2\)
Torsion order: \(1\)
Leading coefficient: \( 4.9676351866091895707546820063874829074 \)
Analytic order of Ш: \( 1 \) (rounded)

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\((i+1)\) \(2\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)
\((-3i-2)\) \(13\) \(2\) \(I_{3}^{*}\) Additive \(1\) \(2\) \(9\) \(3\)
\((2i+3)\) \(13\) \(2\) \(I_{3}^{*}\) Additive \(1\) \(2\) \(9\) \(3\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(3\) 3Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 57122.3-h consists of curves linked by isogenies of degrees dividing 9.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
\(\Q\) 338.f2
\(\Q\) 2704.f2